A Primer on Functional Methods and the Schwinger-Dyson Equations
Eric S. Swanson
TL;DR
This primer introduces functional methods and Schwinger-Dyson equations as a nonperturbative toolkit for quantum field theory, emphasizing practical topics such as diagrammatic generation of high-order SD equations, renormalisation, and numerical solution strategies. It builds from the path integral and functional calculus to the effective action and Legendre transformation, illustrating how 1PI vertices and the effective potential encode nonperturbative physics, including Goldstone's theorem. Concrete applications are presented through scalar $\varphi^4$ theory, fermion contact models, and ladder QED, with detailed discussions of renormalisation, scheme dependence, truncation, and numerical approaches. The notes conclude with guidance on further reading and underscore both the power and the challenges of SD methods, including gauge issues and the importance of robust truncations and computational techniques.
Abstract
An elementary introduction to functional methods and the Schwinger-Dyson equations is presented. Emphasis is placed on practical topics not normally covered in textbooks, such as a diagrammatic method for generating equations at high order, different forms of Schwinger-Dyson equations, renormalisation, and methods for solving Schwinger-Dyson equations.